I found his naturalization record, he enters in 1911 and 1913 but I cannot find him on any normal census yet. Perhaps the American census is similar to the Irish one? In Ireland they enter schools, hospitals etc just by initials, might be worth trying.
If you can somehow private message me your email address I can possibly forward you the images?
In 1911 his address was 141 St. Anns Avenue, NYC He says he is in the State 36 years and was naturalised in 1888 He can read and write and is in good physical condition. By luck, in birthplace of mother, birthplace of father, he puts their names! Edward and Mary Byrne. He is described as able bodied but destitute. He has received relief before from city hosp(ital?) Has not been an inmate in a poorhouse alms house before. Nearest relative, mother 141 St. Anns Avenue, Bronx.
15 May 1913, NYC farm colony Staten Island. He calls himself a waiter and has been admitted due to sickness. There is no discharge date recorded..perhaps he stays here? PArents listed as Edward and Mary Byrne. His address is 4 Rivington Street, NY. His next of kin is brother Francis, also 4 Rivington Street.
I also found Patrick F's passport application with picture if you don't have that.
I would love to hear more about this family!